DLMF:36.2.E6 (Q9841): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1abdec
Property / Symbols used
 
Property / Symbols used: elliptic umbilic canonical integral function / rank
 
Normal rank
Property / Symbols used: elliptic umbilic canonical integral function / qualifier
 
Defining formula:

Ψ ( E ) ⁡ ( 𝐱 ) elliptic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)}}

\ellumbcanonint@{\NVar{\mathbf{x}}}
Property / Symbols used: elliptic umbilic canonical integral function / qualifier
 
xml-id: C36.S2.E5.m3aadec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2abdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3abdec
Property / Symbols used
 
Property / Symbols used: Q12745 / rank
 
Normal rank
Property / Symbols used: Q12745 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q12745 / qualifier
 
xml-id: C36.S1.XMD10.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12746 / rank
 
Normal rank
Property / Symbols used: Q12746 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12746 / qualifier
 
xml-id: C36.S1.XMD11.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12748 / rank
 
Normal rank
Property / Symbols used: Q12748 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12748 / qualifier
 
xml-id: C36.S1.XMD9.m1bdec

Latest revision as of 15:03, 2 January 2020

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English
DLMF:36.2.E6
No description defined

    Statements

    Defining formula
    Ψ ( E ) ⁡ ( 𝐱 ) = 2 ⁢ π / 3 ⁢ exp ⁡ ( i ⁢ ( 4 27 ⁢ z 3 + 1 3 ⁢ x ⁢ z - 1 4 ⁢ π ) ) ⁢ ∫ ∞ ⁢ exp ⁡ ( - 7 ⁢ π ⁢ i / 12 ) ∞ ⁢ exp ⁡ ( π ⁢ i / 12 ) exp ⁡ ( i ⁢ ( u 6 + 2 ⁢ z ⁢ u 4 + ( z 2 + x ) ⁢ u 2 + y 2 12 ⁢ u 2 ) ) ⁢ d u , elliptic-umbilic-canonical-integral 𝐱 2 𝜋 3 𝑖 4 27 superscript 𝑧 3 1 3 𝑥 𝑧 1 4 𝜋 superscript subscript 7 𝜋 𝑖 12 𝜋 𝑖 12 𝑖 superscript 𝑢 6 2 𝑧 superscript 𝑢 4 superscript 𝑧 2 𝑥 superscript 𝑢 2 superscript 𝑦 2 12 superscript 𝑢 2 𝑢 {\displaystyle{\displaystyle\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\sqrt{% \ifrac{\pi}{3}}\,\exp\left(i\left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{% 4}\pi\right)\right)\int_{\infty\exp\left(-7\pi i/12\right)}^{\infty\exp\left(% \pi i/12\right)}\exp\left(i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u% ^{2}}\right)\right)\mathrm{d}u,}}
    0 references
    DLMF id
    DLMF:36.2.E6
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    elliptic umbilic canonical integral function
    Defining formula
    Ψ ( E ) ⁡ ( 𝐱 ) elliptic-umbilic-canonical-integral 𝐱 {\displaystyle{\displaystyle\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)}}
    xml-id
    C36.S2.E5.m3aadec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q12745
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C36.S1.XMD10.m1bdec
    0 references
    Q12746
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C36.S1.XMD11.m1bdec
    0 references
    Q12748
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C36.S1.XMD9.m1bdec
    0 references
     
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